Problem 23 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(3(x+1)=7(x-2)-3\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Simplify the Equation

Apply the distributive law on both sides of the equation to get \(3x + 3 = 7x - 14 - 3\). Afterwards, simplify the right side of the equation to obtain \(3x + 3 = 7x - 17\).

Step 2: Isolate the Variable

Subtract \(3x\) from both sides to get \(3 = 4x - 17\). Now add 17 to both sides to isolate \(x\), resulting in the proposed solution \(x = \frac{20}{4} = 5\).

Step 3: Check the Solution

Substitute \(x = 5\) into the original equation to verify the solution. LHS = \(3(5 + 1) = 18\), RHS = \(7(5 - 2) - 3 = 18\). Since LHS is equal to RHS, the solution is correct.

Key Concepts

Understanding the Distributive Law

In order to solve linear equations, it's essential to understand the distributive law. This law is a property of multiplication that lets us take a term outside of a set of parentheses and distribute it to each term inside the parentheses. In this exercise, you see an example with the equation: \[3(x + 1) = 7(x - 2) - 3\]Using the distributive law here means multiplying 3 by both \(x\) and 1 on the left side, and 7 by both \(x\) and -2 on the right side. This simplifies the equation to:\[3x + 3 = 7x - 14 - 3\]Breaking it down:

• On the left, you distribute 3 to \(x\) getting \(3x\), and to 1 getting 3.
• On the right, 7 is distributed to \(x\) to get \(7x\), and to -2 to get -14. The extra -3 is a constant term.

Understanding this step is crucial as it simplifies complex equations and helps in further solving the equation.

Isolating the Variable

The next step in solving the equation is to isolate the variable, which is crucial for finding the exact value of \(x\). This involves getting \(x\) alone on one side of the equation. Here's how it's done in this problem:We have the simplified equation:\[3x + 3 = 7x - 17\]To isolate \(x\), you need to:

• Subtract \(3x\) from both sides to remove the \(x\) term from the left side. This gives us: \[3 = 4x - 17\]
• Add 17 to both sides to get rid of the negative constant from the right side: \[3 + 17 = 4x\]
• Finally, divide by 4 to solve for \(x\): \[x = \frac{20}{4} = 5\]

By continuously simplifying and rearranging, we isolate \(x\) to find its value, which is 5. The key is to perform the same operations on both sides of the equation to maintain equality.

Checking Your Solutions

Once you have a proposed solution for your variable, it’s vital to check your work. This ensures that your solution is indeed correct by substituting the solved value back into the original equation. Here's how you do it with our problem:We found that \(x = 5\). Now, substitute 5 back into the original equation:\[3(x + 1) = 7(x - 2) - 3\]Substitute 5 for \(x\):

• Left-hand side (LHS): \[3(5 + 1) = 3 \cdot 6 = 18\]
• Right-hand side (RHS): \[7(5 - 2) - 3 = 7 \cdot 3 - 3 = 21 - 3 = 18\]

Both sides equaling 18 confirms that our solution \(x = 5\) is correct. Always remember to perform this step as a self-check for accuracy.